Hadamard Matrix/Examples/Arbitrary 2 x 2
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Example of Hadamard Matrix
An arbitrary example of a $2 \times 2$ Hadamard matrix is:
- $\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}$
Proof
We multiply $\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}$ by its transpose, and find:
- $\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix} \begin {pmatrix} 1 & -1 \\ 1 & 1 \end {pmatrix} = \begin {pmatrix} 2 & 0 \\ 0 & 2 \end {pmatrix} = 2 \begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$
Hence the result by definition of Hadamard matrix.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hadamard matrix
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hadamard matrix